On automorphisms of extremal even unimodular lattices of dimension 48

TL;DR

This paper investigates the automorphism groups of extremal even unimodular lattices in dimensions 48 and 72, providing classifications and restrictions on automorphisms, and identifying unique lattice structures related to cyclotomic fields.

Contribution

It computes automorphism groups of known extremal lattices and classifies all 48-dimensional extremal lattices with certain automorphism properties, revealing unique lattice structures.

Findings

Automorphism groups of three 48-dimensional and one 72-dimensional lattices are computed.

Restrictions on possible automorphisms of 48-dimensional extremal lattices are established.

The lattice P_{48n} is uniquely characterized as an ideal lattice over a cyclotomic field.

Abstract

The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are computed using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order $m$ with $\varphi(m) > 24$. In particular the lattice $P_{48n}$ is the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.